Investigation 3: Markups, Markdowns, and Measures: Using Ratios, Percents, and Proportions

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1 Comparing and Scaling: Ratios, Rates, Percents & Proportions Name: Per: Investigation 3: Markups, Markdowns, and Measures: Using Ratios, Percents, and Proportions Standards: 7.RP.1: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. 7.RP.2a: Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane (observing whether the graph is a straight line through the origin). 7.RP.2c: Represent proportional relationships by equations. 7.RP.2d: Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. 7.RP.3: Use proportional relationships to solve multistep ratio and percent problems. Date Learning Target/s Classwork (Check Off Completed/ Monday, Jan. 4 Tuesday, Jan. 5 Wednesday, Jan. 6 Thursday, Jan. 7 Friday, Jan. 8 Monday, Jan. 11 I can complete rate tables, find unit rates, and write/graph equations for proportional relationships. I can use proportional relationships to solve multistep % problems. I can use proportional relationships to solve multistep % problems. I can use proportional relationships to solve conversion problems. I can review for the unit test and make a study plan for myself based on my strengths/challenge areas. Corrected Items) Pg. 2-3: Inv. 2 Review Record Scores for Check Up 2 Pg. 5-7: Inv. 3.1 Proportions with % Pg. 9-11: Inv. 3.1, Part 2 Exit Ticket Pg : Inv. 3.2 Begin Unit Review Correct Unit Review Homework (Check Off Completed/ Corrected Items) Pg. 4: Correct with Zaption CS 2 Review Pg. 8: Correct with Zaption CS 3.1 Percents Pg. 12: Zaption CS 3.2 Notes for Conversions Complete Inv. 3 Packet (All Pages) Packet Signature Complete Unit Self-Assess Your Understanding of the Learning Target/s Review Google Doc Unit Review Tuesday, Jan. 12 I can show my mastery of unit standards. Comparing and Scaling Unit Test MSA Unit Readiness You ve got this! Check Up 2 I can compare ratios in real-world situations. Score: / 6 I can find the unit rate and the constant of proportionality. Score: / 9 I can determine whether two quantities are in a proportional relationship. Exit Ticket 3 I can solve multistep ratio and percent problems. (7.RP.3) Score: / Parent/Guardian Signature: Due: 1

2 Review of Inv. 2 Rate Tables, Unit Rates, Equations, and Graphs of Proportional Relationships A. Which of these items is the better buy: a 40-pack of pencil-top erasers for $2.82 or a 2-pack of pencil-top erasers for $0.12? Show your thinking. B. For each situation, complete the rate table, find two unit rates, write two equations, and create two graphs. a. 3 dozen apples for $4.50 Rate Table: Apples (dozen) 3 1 Cost ($) $4.50 $1 Show how you completed the rate table by setting up and solving proportions (find a scale factor): 3 dozen apples = 1 dozen apples 3 dozen apples = $4.50 $4.50 $1 Unit Rates (include labels): per 1 dozen apples per $1 Equations: c = cost, d = dozens of apples c = d d = c Circle the constant of proportionality in each equation. Graphs (include labels on axes): Equation: c = d Equation: d = c 2

3 b. 30 bottles of water for $4.80 Rate Table: Water bottles 30 1 Cost ($) $4.80 $1 Show how you completed the rate table by setting up and solving proportions (find a scale factor): 30 water bottles = 30 water bottles = $4.80 $4.80 Unit Rates (include labels): per per Equations: w = water bottles, c = cost w = c = Circle the constant of proportionality in each equation. Graphs (include labels on axes): Equation: w = Equation: c = Put a star on the point (1, ) on each graph. What does that point represent on each graph? 3

4 Homework: Complete and then correct with Zaption CS 2 Review Complete the rate table, find two unit rates, write two equations, and create two graphs. 24 ounces of mozzarella cheese for $2.88 Rate Table: Mozzarella (ounces) 24 1 Cost ($) $2.88 $1 Show how you completed the rate table by setting up and solving proportions (find a scale factor): = = Unit Rates (include labels): Equations: m = mozzarella, c = cost Circle the constant of proportionality in each equation. Graphs (include labels on axes): Equation: Equation: Put a star on the point (1, ) on each graph. What does that point represent on each graph? 4

5 Inv. 3.1: Commissions, Markups, and Discounts Proportions with Percents Determine Whether Two Quantities are in a Proportional Relationship (7.RP.2a) 1. Circle ALL the equations below that show a proportional relationship between x and y. A. y = 6x B. y = x 6 C. y = 3x + 2 D. y = x 2 2. Circle ALL the tables below that represent a proportional relationship between x and y. 3. Circle ALL the graphs below that show a proportional relationship between x and y. A. B. C. D. Summary: How can you tell if an equation, table, or graph is a proportional relationship? Equation Table Graph 5

6 Salespeople who sell cars, houses, and fancy jewelry often work on commission. Typically a commission is a of the sale price of an item. Alternatively, a commission may be a percentage of the item s markup. The markup is the between the buying price, the cost for a store or dealer to buy an item, and the selling price, the price the store or dealer sets for their customers. To make a profit, the selling price must be higher than the buying price. Huan sells cars at Carla s Used Cars. A. At Carla s Used Cars, Huan earns a commission that is 25% of the markup on the car. Huan recently sold a sedan ($300 markup), SUV ($500 markup), and convertible ($950 markup). Car Markup Commission Buying Price Selling Price Sedan $300 SUV $500 Convertible $ For each car, what was Huan s commission in dollars? Add each value to the table above and then explain how you found the commissions. 2. At Carla s, the markup on a car is 10% of the buying price, the price at which Carla bought the car. a. For each car, what was the buying price? Add each value to the table above and then explain how you found the buying prices. b. For each car, what was the selling price? Add each value to the table above and then explain how you found the selling prices. 6

7 3. Carla buys a minivan for $20,500. She writes a proportion to find the selling price, S. Is Carla s method correct? Explain. S = 20, Add the value of the selling price to the table below. Car Markup Commission Buying Price Selling Price Minivan $20, Huan checks the selling price Carla found. He uses M to represent the markup. Is Huan s method correct? Explain. M = 20, Add the value of the markup to the table. If he wants to use the value of the markup to find the selling price, what does he need to do? 5. How much commission would Huan make from selling the minivan? (He earns 25% of the markup.) Set up and solve a proportion with percents to prove your answer. = Add the value of the commission to the table. 6. Challenge: A customer wants to buy the minivan. Her budget is $23,000. The selling price plus 5% sales tax goes over the customer s budget. What maximum selling price can the customer afford? Explain. 7

8 Homework: Complete and then correct with Zaption CS 3.1 A. Recently Huan received a job offer from Otto s Used Autos. At Otto s, the markup is 15% of the buying price. The commission at Otto s is 20% of the markup. Huan wonders whether he will get higher commissions at Carla s or at Otto s. 1. Otto also bought a minivan for $20,500. What would be the markup on the minivan? 2. What would be the commission on the minivan? 3. Should Huan work for Carla or for Otto? Explain. (Hint: Look at yesterday s class work to find the commission he would make on the minivan if he worked for Carla.) B. Bill s Bikes sells new and used bikes. Bill buys used bikes, fixes them, and marks up the prices by 80%. The saleperson selling the bike gets a 25% commission on the markup. 1. Find the missing values in the table. Costs and Revenue for Roberto s Sales Buying Price Markup (80% of buying price) Selling Price Commission (25% of markup) Profit (money the shop makes on the sale) $100 $80 $180 $20 $60 $10 $55 $125 8

9 Inv. 3.1: Commissions, Markups, and Discounts Proportions with Percents, Part 2 A. Jackson uses 1 cup of cranberry juice for every 1 cup of raspberry juice to make a fruit drink. How many cups of 8 4 cranberry juice will he need to use for 1 1 cups of raspberry juice? Express your answer as a fraction. 2 B. Mrs. Buck has a 16-ounce mug filled with coffee. She drinks 9 ounces. What is the percentage of ounces that she has left in her mug? Round your answer to the nearest tenth. C. Identify which estimate seems to the most reasonable: 1. 5% tax on a $42 purchase: Under $2 Exactly $2 Over $2 2. 9% tax on a $59.99 purchase: Under $6 Exactly $6 Over $6 D. Find the sales tax for each item. Set up and solve proportions. 1. A sweater for $36 at 7% sales tax = 2. A skateboard for $62.80 at 6% sales tax = 9

10 E. The ratio of Bill s Bikes sells new and used bikes. Bill buys used bikes, fixes them, and marks up the prices by 80%. The saleperson selling the bike gets a 25% commission on the markup. 1. Find the missing values in the table. Costs and Revenue for Linda s Sales Buying Price Markup (80% of buying price) $48 Selling Price Commission (25% of markup) Profit (money the shop makes on the sale) $252 $14.40 $54 $N Show all of your work below: When you have completed the table, continue to the next page 10

11 F. For each arrow in the figure at right, write a mathematical rule describing how to get from one value to the next value. The first one is done for you. (Use your work from the page before to help you.) G. For each part, write two equations for the listed relationship. (Use the figure above to help you.) 1. The markup amount and the buying price markup amount = buying price = 2. The buying price and the selling price buying price = selling price = 3. The commission and the markup amount = = 4. The profit and the commission 11

12 Homework: Use Zaption CS 3.2: Notes for Conversions to complete the notes below. You can use unit rates, such as 3 feet per yard, to convert measurements. The following relationships may be helpful in our next class work problems. You can also use proportions to convert measurements. For example: = A. Kate walks 5 miles in 2 hours at a steady pace. How far can she walk in 1 hour and 15 minutes? Strategy: Set up and solve a proportion with a scale factor: = 1 hour and 15 minutes = hours B. There are 276 calories in 6 ounces of chicken. How many calories are in 1 pound of chicken? Strategy: Find a unit rate and scale up with a scale factor: = = 1 pound = ounces C. One cup of whole milk has 8 grams of fat. How many grams of fat are in a gallon of whole milk? Strategy: Use a rate table, find a unit rate, and scale up with a scale factor: 1 gallon = cups Grams of Fat Cups of Milk D. Choose your own strategy and solve the problem below: Nathan s lawnmowers use 2/3 of a tank of gas to cut three one-acre lawns, How many one-acre lawns can he cut with a full tank of gas? 12

13 Inv. 3.2: Measuring to the Unit Measurement Conversions Choose a strategy to solve each problem ratios, proportions, unit rates, rate tables, equations, etc.: A. Chetan makes a necklace for his sister. Twelve beads take up 5 inches of strong. How many beads fit on 1 foot of string? B. Chetan wants to make a necklace with 50 beads. He knows that 12 beads take up 5 inches of string, but the store only sells string by the centimeter. How many centimeters should he buy? (Note: 1 inch = 2.5 centimeters) C. Sean walks ¾ of a mile in 15 minutes. At the same pace, how far can Sean walk in 1 hour and 20 minutes? D. Can you make sense of these strategies for solving problem C? a. Sean writes the expression: 3 1 and completes the division. What information does this expression 4 4 give Sean? b. Davina tells Sean to use a proportion to find how far he can walk in 1 hour and 20 minutes. She writes: How is Davina s strategy similar to Sean s? How is it different? c. How is Sean s strategy similar to using a rate table? How is it different? 13

14 E. Allen runs 8 miles in 3 hours at a steady pace. How long does it take him to run 3 miles? (Give your answer in minutes.) F. Maren walks 3/5 mile in 24 minutes at a steady pace. How long does it take her to walk 2 miles? (Give your answer in minutes.) G. Half an avocado has about 160 calories. How many calories do a dozen avocados have? H. There are about 1.5 grams of fat in 1 tablespoon of hummus. How many grams of fat are in 2.5 cups of hummus? (Note: 16 tablespoons = 1 cup) I. How many ounces are in 10 ½ pounds? J. How many cups are in 55 gallons? K. For each problem below, describe what value x represents and then solve for x. L. Challenge: Solve each proportion for x: 14

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